30 September 2009

Is zero even?

Did you know that there are actually things to say about whether zero is even or odd (from Wikipedia)? Obviously it is, but the math-ed folks have seriously looked at this.

I found this via a comment by John Thacker at The Volokh Conspiracy. There's a poll there; right now 2% of people have said 0 is odd, 51% even, 43% both, 4% neither. I can kind of understand what's going on with people saying "neither" (perhaps they're getting this from some elementary-school notions), but how is 0 odd?

My answer: yes, zero is even, because it's twice an integer.

(Or because the identity permutation on n letters is an element of the alternating group An -- I've been thinking about permutations a lot lately. But if you understand that, you probably are like me, think zero is even, and didn't even think there was anything to discuss.)

Incidentally, sometime recently -- I forget the context -- I saw something that referred to the Gaussian integer a+bi as "uneven" if and only if a and b had different parity.

24 comments:

Anonymous Rex
said...

The sign of a permutation is actually pretty confusing: odd cycles are even, while even cycles are odd.

The comments on the Volokh Conspiracy are a goat rope. My favorite comment was "It would be odd if it were not even." A great find was the book McGraw-Hill’s Catholic High School Entrance Exams, 2ed. (I particularly enjoyed the line "even integer +/- even integer = even integer" just after defining 0 as not even.)

A book by David J. Benson (a mathematician) says at some point that "The left side is even and the right side is odd, so their common value is both even and odd, and hence zero." which would be frightening if he weren't talking about functions!

Wow, I'm really surprised to see how much debate there is. It seems so obvious to me!

@Carl: we say 1 isn't prime just so the theorems are shorter. Otherwise, loads of them woud be "for every prime number larger than 1..." and the expression "prime larger than 1" would be used many times more often that the word "prime" alone.

Zero being odd doesn't cause any problems like one being prime would. On the contrary, if we didn't consider 0 even, we'd often have to say things like "all even numbers and zero".

I was also surprised to discover last year that there is so much uncertainty about this. My four-year-old asked me whether it was even or odd. Our formula for evenness has followed the example of "Is 14 even?" "Yes, because it is 7+7." "Is 13 even?" "No, because it is 6+6+1". So when she asked if zero was even I said of course, because it is 0+0. But the other adults in the room looked at me strangely and asked if I was sure. Fortunately this is an issue on which I don't have to hedge, or simplify; I just said yes, zero is even.

I just asked out company's illustrator, "is zero even or odd", and he said "can't it be neutral?" He had no difficulty identifying 2 as even. When I pointed out that, in fact, zero is even, he said, "you see, I was right".

I'm still trying to wrap my mind around an understanding of why someone would think that zero is anything other than the most even of all the even numbers. If you're going to be unsure about a number being even or odd, it should be something large like:7^193161 + 17163131341663(which is even).

Maybe it has something to do with the absence of a multiplicative inverse of zero. But I suspect that instead it has to do with zero and one being "special" and not included in the primes and composites.

I asked Iris (age 5.3) today to see what she would say, and she said it was even. When I asked why, she said it was because 1 was odd, and even and odd numbers always alternate.

Usually in such cases I likes to probe a little deeper, perhaps by bringing up some complication that confuses the issue. "But what about xyz," one says. "Shouldn't that mean that 0 is odd?" But in this case I could think of no such argument.

I guess that's what's puzzling about the whole issue. None of us seems to have any good ideas about why anyone would think otherwise.

The question of whether or not 1 is prime actually makes a certain amount of sense as a debate. I must admit that I couldn't think of a good reason why it shouldn't be prime until I discovered finite fields.

unapologetic, it's presumably because for any prime, there is a field with that number of elements. But there is no field with 1 element because you need to have 0 and 1, so the least size is 2.

'"But what about xyz," one says. "Shouldn't that mean that 0 is odd?" But in this case I could think of no such argument.'

This is exactly right. I can think of no reason why 0 should be odd.

However, here's an argument: Human use of math is based on the desire for justice, hence our concentration on equality. If you have an even number of cupcakes, you can divide them equally so that Alice and Bob get the same number. But with zero cupcakes, there's nothing to divide.

Carl, that's exactly what I was hoping someone would say. And then I could point out that "but the only reason there's no 'field with 1 element' is that we make an arbitrary insistence that the additive and multiplicative identities of a ring be distinct".

In fact, there are many, many reasons why the field with one elementshould exist. Just because the undergraduate notion of sets-with-structure doesn't capture it doesn't mean it's not there.

unapologetic, in a sense, I guess you could say that the axiom that 1 is not prime is the same as the axiom that in all fields, 0≠1, or any equivalent axiom (e.g. the axiom that in any field, x/0 is not defined for any x).

But this is beside the point.

Caution: Nonstandard terminology follows.

Let p be a prime. Define the Galois field of order p GF(p) as the ring of p-adic integers with a suitable division operation.

If 1 were a prime, then it would not be the case that GF(p) is a field for all prime p. Even if you agree that there is a finite field with one element, it's not a Galois field.

Then the axiom that 0≠1 merely implies that all finite fields are Galois fields and vice versa. Or, to put it another way, if you do not admit the axiom 0≠1 (or equivalent), then all this would show is that there is a finite field which is not a Galois field.

And finally, all I said is that this was the first comprehensive argument that I encountered as to why 1 should not be placed in the category "prime number". This is an extremely modest claim.

Interesting. Remember my mum teaching her primary school kids about rounding numbers to the nearest hundred. There was a bit of debate about how to round the number 30. Mum thought the answer should be 100 because "zero isn't a hundred, so the nearest hundred is 100".

plutoman, I found that comment interesting because rounding is a more difficult problem than most people appreciate. How you round depends on what kind of error you're trying to minimise (absolute vs relative error, while also avoiding systematic error).

If you had to round 40 to the nearest power of 10, you should pick 1 if you're minimising absolute error, and 100 if you're minimising relative error.

unapologetic, sorry, I made a couple of mistakes there; I blame lack of caffeine.

You should round 40 to 100 rather than 10 (I incorrectly said 1) to minimise the relative difference, not relative error. The relative difference between 10 and 40 is (40-10)/40 = 0.75, and the relative difference between 40 and 100 is (100-40)/100 = 0.6.

I think 0 being odd comes if you don't think it's even. Zero is a strange number, so 0=0+0 doesn't feel like "proof" that it's even to everyone [you can put 0 objects in two piles of 0, but that just feels weird and doesn't fly with everyone].

If your definition of odd is "not even" rather than something along the lines of 2K+1, then if 0 isn't even it must be odd.

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The Universe includes everything that exists. In the Universe there are billions and billions of stars. These stars are distributed in the space in huge clusters. They are held together by gravitation and are known as galaxies. Sun is also a star. Various members of the solar system are bound to it by gravitation force. Gravitation force is the ultimate cause of birth and death of galaxy, star and planets etc. Gravitation can be considered as the cause of various forms of animate and inanimate existence. Human form is superior to all other forms. Withdrawal of gravitational wave from some plane of action is called the death of that form. It can be assumed that gravitation force is ultimate creator. Source of it is ‘God’. Gravitational Field is the supreme soul (consciousness) and its innumerable points of action may be called as individual soul (consciousness). It acts through body and mind. Body is physical entity. Mind can be defined as the function of autonomic nervous system. Electromagnetic waves are its agents through which it works. This can be realized through the practice of meditation and yoga under qualified meditation instruction. This can remove misunderstanding between science and religion and amongst various religions. This is the gist of all religious teachings – past, present and future.

In my view, in scientific terminology source of gravitational wave is God.